Deformable Contour Tracking using PF-MT (Particle Filter with Mode Tracker)
(collaborators: Anthony Yezzi, Yogesh Rathi and Allen Tannenbaum at Georgia Tech)
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We consider the problem of tracking the boundary contour of a moving and deforming object from a sequence of images. If the motion of the “object” or region of interest is constrained (e.g. rigid or approximately rigid), the contour motion can be efficiently represented by a small number of parameters, e.g. the affine group. But if the “object” is arbitrarily deforming, each contour point can move independently. Contour deformation then forms an infinite (in practice, very large), dimensional space. Direct application of particle filters for large dimensional problems is impractical, due to the reduction in effective particle size as dimension increases. But in most real problems, at any given time, “most of the contour deformation” occurs in a small number of dimensions (“effective basis”) while the residual deformation in the rest of the state space (“residual space”) is “small”. The effective basis may be fixed or time varying. Based on this assumption, we modify the particle filtering method to perform sequential importance sampling only on the effective basis dimensions, while replacing it with deterministic mode tracking in residual space (PF-MT). We develop the PF-MT idea for contour tracking.
Deforming contours occur either due to changing region of partial occlusions or when the object of interest is actually deforming its shape over a time or space sequence of images. Examples of the second kind are a beating heart, moving animals or humans, or the cross-sections of different parts of a 3D object like the brain, in consecutive MRI slices. Most biological images contain deforming objects/regions. Contour tracking has many applications in medical image analysis, e.g. sequential segmentation of volume images; tracking heart regions or image guided surgery. The observation likelihood is often multimodal due to background objects (clutter) which are partially occluded by the “object of interest” or due to an object which partially occludes the “object of interest” or due to low contrast imagery. Heavy tailed and often multimodal observation likelihoods occur when the observation noise has occasional outliers.
In our initial work (CVPR 2005, PAMI 2007), we treated the 6 dimensional space of affine deformations as the "effective basis" while the space of non-affine deformation was the residual space. The implicit assumption is that the posterior of non-affine deformation (conditioned on affine deformation and the current image) is unimodal. This is valid for many practical problems where the non-affine deformation per frame is small, e.g. a rigid object tracked by a perspective camera with frequent viewpoint changes, or approximately rigid objects, e.g. human body contour from a distance.
But in other situations, where local deformations are large, there may be more than one non-affine mode for the same affine deformation value and the same image, i.e. posterior of non-affine deformation may be multimodal. Example applications are a rigid object undergoing partial occlusions, e.g. a car going under a light pole, or tracking regions of interest in low contrast medical images (multiple nearby contour modes due to the low contrast). Such applications also require importance sampling on the space of non-affine deformations. In recent work (CDC 2006, Trans. IP, Accepted 2008), we use global translations and deformation velocity at subsampled contour locations interpolated using a B-spline basis as the effective basis. The effective basis dimension is allowed to change with time. We are able to get excellent results with as low as K=6 subsampled points which is much smaller than the total number of contour points, M=150-200. Or in other words, the deformation “signal” is approximately bandlimited (spatially), with the approximate cut-off frequency being much smaller than the maximum measurable frequency, 0.5Hz (cycles/pixel). We can increase K if the approximate cut-off frequency increases.